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For quite some time I have been wondering how automatic differentiation works. However, I am a bit confused on how the forward mode works -- I am not equipped to deal with reverse mode at the moment. I have tried to read the source code of some libraries (mainly autodiff) and read some papers (e.g. FAD) in order to understand how people are doing it, with little success.
My main issue is I don't get how dual numbers are used. For example, let's say we define a class of dual numbers (in C++) that holds two numbers; value and derivative. Then, we can overload different mathematical functions and operators, in order to define the dual number algebra (as in the complex number case). Then, and this is my problem, no matter we do, we are only going to get first derivatives.
I keep reading about implementation of hyper-dual numbers, which are described as duals that store values, Jacobian, Hessian, etc. If this is true, then if I have a function of 15 variables and I need the third derivative wrt all of them, my computer is going to blow up... Since there are very efficient libraries out there that do such calculations, I am clearly missing something.
I don't have a specific coding question, I would appreciate any input on how forward mode autodiff can be implemented in a practical way.
More info
I have written a basic dual number library in C++, which you can find on github. However, once I finished writing the class and a few function overloads, I gave up due to the problem I describe above (DualNumbers.cpp has several examples, thogouh).
Recently I also started again, this time using expression templates (because I wanted to learn how to use them) -- see github, but this approach has another issue I describe in another question.
Native support for differential programming has been added to Swift for the Swift for Tensorflow project. Julia has similar with Zygote.
What exactly is differentiable programming?
what does it enable? Wikipedia says
the programs can be differentiated throughout
but what does that mean?
how would one use it (e.g. a simple example)?
and how does it relate to automatic differentiation (the two seem conflated a lot of the time)?
I like to think about this question in terms of user-facing features (differentiable programming) vs implementation details (automatic differentiation).
From a user's perspective:
"Differentiable programming" is APIs for differentiation. An example is a def gradient(f) higher-order function for computing the gradient of f. These APIs may be first-class language features, or implemented in and provided by libraries.
"Automatic differentiation" is an implementation detail for automatically computing derivative functions. There are many techniques (e.g. source code transformation, operator overloading) and multiple modes (e.g. forward-mode, reverse-mode).
Explained in code:
def f(x):
return x * x * x
∇f = gradient(f)
print(∇f(4)) # 48.0
# Using the `gradient` API:
# ▶ differentiable programming.
# How `gradient` works to compute the gradient of `f`:
# ▶ automatic differentiation.
I never heard the term "differentiable programming" before reading your question, but having used the concepts noted in your references, both from the side of creating code to solve a derivative with Symbolic differentiation and with Automatic differentiation and having written interpreters and compilers, to me this just means that they have made the ability to calculate the numeric value of the derivative of a function easier. I don't know if they made it a First-class citizen, but the new way doesn't require the use of a function/method call; it is done with syntax and the compiler/interpreter hides the translation into calls.
If you look at the Zygote example it clearly shows the use of prime notation
julia> f(10), f'(10)
Most seasoned programmers would guess what I just noted because there was not a research paper explaining it. In other words it is just that obvious.
Another way to think about it is that if you have ever tried to calculate a derivative in a programming language you know how hard it can be at times and then ask yourself why don't they (the language designers and programmers) just add it into the language. In these cases they did.
What surprises me is how long it to took before derivatives became available via syntax instead of calls, but if you have ever worked with scientific code or coded neural networks at at that level then you will understand why this is a concept that is being touted as something of value.
Also I would not view this as another programming paradigm, but I am sure it will be added to the list.
How does it relate to automatic differentiation (the two seem conflated a lot of the time)?
In both cases that you referenced, they use automatic differentiation to calculate the derivative instead of using symbolic differentiation. I do not view differentiable programming and automatic differentiation as being two distinct sets, but instead that differentiable programming has a means of being implemented and the way they chose was to use automatic differentiation, they could have chose symbolic differentiation or some other means.
It seems you are trying to read more into what differential programming is than it really is. It is not a new way of programming, but just a nice feature added for doing derivatives.
Perhaps if they named it differentiable syntax it might have been more clear. The use of the word programming gives it more panache than I think it deserves.
EDIT
After skimming Swift Differentiable Programming Mega-Proposal and trying to compare that with the Julia example using Zygote, I would have to modify the answer into parts that talk about Zygote and then switch gears to talk about Swift. They each took a different path, but the commonality and bottom line is that the languages know something about differentiation which makes the job of coding them easier and hopefully produces less errors.
About the Wikipedia quote that
the programs can be differentiated throughout
At first reading it seems nonsense or at least lacks enough detail to understand it in context which is why I am sure you asked.
In having many years of digging into what others are trying to communicate, one learns that unless the source has been peer reviewed to take it with a grain of salt, and unless it is absolutely necessary to understand, then just ignore it. In this case if you ignore the sentence most of what your reference makes sense. However I take it that you want an answer, so let's try and figure out what it means.
The key word that has me perplexed is throughout, but since you note the statement came from Wikipedia and in Wikipedia they give three references for the statement, a search of the word throughout appears only in one
∂P: A Differentiable Programming System to Bridge Machine Learning and Scientific Computing
Thus, since our ∂P system does not require primitives to handle new
types, this means that almost all functions and types defined
throughout the language are automatically supported by Zygote, and
users can easily accelerate specific functions as they deem necessary.
So my take on this is that by going back to the source, e.g. the paper, you can better understand how that percolated up into Wikipedia, but it seems that the meaning was lost along the way.
In this case if you really want to know the meaning of that statement you should ask on the Wikipedia talk page and ask the author of the statement directly.
Also note that the paper referenced is not peer reviewed, so the statements in there may not have any meaning amongst peers at present. As I said, I would just ignore it and get on with writing wonderful code.
You can guess its definition by application of differentiability.
It's been used for optimization i.e. to calculate minimum value or maximum value
Many of these problems can be solved by finding the appropriate function and then using techniques to find the maximum or the minimum value required.
I have good knowledge of C++ but never dwell into STL. What part of STL I must learn to improve productivity and reduce defects in my work?
Thanks.
I have good knowledge of C++
With all due respect, but no – you don’t. The standard library, or at least large parts of it (especially the subset known as “STL”) is a fundamental part of C++. Without knowledge of it you don’t know very much about C++ at all.
In fact, much of the modern design of C++ (essentially everything since the 98 version) was guided by design considerations stemming from the standard library, and much of the changes in the language since then are changes to the standard library. If you take a look at the official C++ language description a good part of the document is concerned with the library.
Usually the first reaction (at least in my opinion, of course) for people who have not worked with the STL before is to get upset with all the template code. So I would start by studying a little bit on this subject.
In the case you already know template fundamentals I would recommend taking a brief look over an STL design document. This is actually the second stage of hassle for people not yet familiar with it. The reason for this is that the STL is not designed under a typical object oriented paradigm, but under the generic programming paradigm.
With this in mind, a good start could be this introductory article. The terms used throughout the STL components are explained there. Please notice that is a relatively old text focused on the SGI implementation (which predates the C++ standard and incorrectly mentions, for example, the hash based containers as part of it). However, the theory is still valid.
Well, if you already know most things I've said up to this point, just jump directly to the topcis the others provided.
You mention about improving your productivity and reduce defects. There are general guidelines that you can use for this. I assume c++11, and I mention a bit more than stl (smart pointers):
Use containers, they will manage memory for you. You get rid of new for C arrays and later having to delete them, for example.
For dynamic arrays, use std::vector. You also have hashtables in std::unordered_map and balanced trees with std::map. There are more containers, take a look here.
Use std::array instead of plain C arrays whenever you can: they never decay to pointers when passing as arguments to functions, which can avoid very disgusting bugs.
Use smart pointers and forget forever for a naked new and its matching delete in code.
This can reduce errors far more than you would expect, especially in the presence of exceptions.
Use std::make_shared when possible. You can use it to allocate a shared_ptr directly as an argument to a function that takes a std::shared_ptr. With a naked new this is not possible.
Use algorithms instead of hand-coded loops. The code will be far more readable and usually more performant.
With this advice your code should look closer (but not necessarily equal or semantically equivalent) to C# or Java, in which manual memory management disappears. This is even better than garbage collection in many scenarios, since you have deterministic guarantees for when a resource will be freed.
I'd say the algorithms from <algorithm> will really clean up your code and at the same time make your code more concise.
Obviously, knowledge of all the containers will help you to optimize the bottlenecks of your code caused by a certain choice of container which is not optimal (but be sure to profile first).
These are pretty much the basics and they will help you a lot to make more robust code.
After that you can delve into smart pointers like std::shared_ptr which are almost always better than regular pointers (in my case, at least).
I think can start with containers(vector, list) and alghorithms(binary search, sort).
And as Jesse Emond wrote, the more you know, the better you live)))
I remember when I was in DSA I was like wtf O(n) and wondering where would I use it other than in grad school or if you're not a PhD like Bloch. Somehow uses for it does pop up in business analysis, so I was wondering when have you guys had to call up your Big O skills to see how to write an algorithm, which data structure did you use to fit or whether you had to actually create a new ds (like your own implementation of a splay tree or trie).
Understanding Data Structures has been fundamental to many of the projects I've worked on, and that goes beyond the ten minute song 'n dance one does when asked such a question in an interview situation.
Granted that modern environments with all sorts of collection classes can make light work of storing and accessing large amounts of data, but having an understanding that a particular problem is best solved with a particular data structure can be a great timesaver. And by "timesaver" I mean "the difference between something working and not working".
Honestly, being able to answer that stuff is my biggest criterion for taking interviewees seriously in an interview. Knowing how basic data structures work, basic O(n) analysis, and some light theory is really crucial to being able to write large applications successfully.
It's important in the interview because it's important in the job. I've worked with techs in the past that were self taught, without taking the data structures course or reading a data structures book, and their code is occasionally bad in ways they should have seen coming.
If you don't know that n2 is going to run slowly compared to n log n, you've got more to learn.
As far as the later half of the data structures courses, it isn't generally applicable to most tech jobs, but if you ever do wind up needing it, you'll wish you had paid more attention.
Big-O notation is one of the basic notations used when describing algorithms implemented by a particular library. For example, all documentation on STL that I've seen describes various operations in terms of big-O, so naturally you have to e.g. understand the difference between O(1), O(log n) and O(n) to understand the implications of your choice of STL containers and algorithms. MSDN also does that for .NET classes, and IIRC Java documentation does that for standard Java classes. So, I'd say that knowing the notation is pretty much a requirement for understanding documentation of most popular frameworks out there.
Sure (even though I'm a humble MS in EE -- no PhD, no CS, differently from my colleague Joshua Block), I write a lot of stuff that needs to be highly scalable (or components that may need to be reused in highly scalable apps), so big-O considerations are most always at work in my design (and it's not hard to take them into account). The data structures I use are almost always from Python's simple but rich supply (which I did lend a hand developing;-), rarely is a totally custom one needed (rather than building on top of list, dict, etc); but when it does happen (e.g. the bitvectors in my open source project gmpy), no big deal.
I was able to use B-Trees right when I learned about them in algorithm class (that was about 15 years ago when there were much less open source implementations available). And even later the knowledge about the differences of e. g. container classes came in handy...
Absolutely: even though stacks, queues, etc. are pretty straightforward, it helps to have been introduced to them in a disciplined fashion.
B-Tree's and more advanced sorting are a bit more difficult so learning them early was a big benefit and I have indeed had to implement each of them at various points.
Finally, I created an algorithm for single-connected components a few years back that was significantly better than the one our signal-processing team was using but I couldn't convince them that it was better until I could show that it was O(n) complexity rather than O(nlogn).
...just to name a few examples.
Of course, if you are content to remain a CRUD-system hacker with no real desire to do more than collect a paycheck, then it may not be necessary...
I found my knowledge of data structures very useful when I needed to implement a customizable event-driven system about ten years ago. That's the biggie, but I use that sort of knowledge fairly frequently in lesser ways.
For me, knowing the exact algorithms has been... nice as background knowledge. However, the thing that's been the most useful is the more general background of having to pay attention to how different pieces of an algorithm interact. For instance, there can be places in code where moving one piece of code (ie, outside a loop) can make a huge difference in both time and space.
Its less of the specific knowledge the course taught and, rather, more that it acted like several years of experience. The course took something that might take years to encounter (have drilled into you) all the variations of in pure "real world experience" and condensed it.
The title of your question asks about data structures and algorithms, but the body of your question focuses on complexity analysis, so I'll focus on that too:
There are lots of programming jobs where being able to do complexity analysis is at least occasionally useful. See What career can I hope for if I like algorithms? for some examples of these.
I can think of several instances in my career where either I or a co-worker have discovered a a piece of code where the (usually time, sometimes space) complexity was higher that it should have been. eg: something that was quadratic or cubic when it could have been linear or nlog(n). Such code would work fine when given small inputs, but on larger inputs would quickly become really slow or consume all available memory. Knowing alternative algorithms and data structures, their complexities, and also how to analyze the complexity to build new algorithms is vital in being able to correct these problems (or avoid them in the first place).
Networking is all I've used it: in an implementation of traveling salesman.
Unfortunately I do a lot of "line of business" and "forms over data" apps, so most problems I work on can be solved by hammering together arrays, linked lists, and hash tables. However, I've had the chance to work my data structures magic here and there:
Due to weird complex business rules, I worked on an application which used a custom thread pool implemented as a leftist-heap.
My dev team struggled to write a complex multithreaded app. It was plagued with race conditions, dead locks, and lousy performance due to very fine-grained locking. We re-worked the code to share state between threads, opting to write a very light-weight wrapper to facilitate message passing. Next, we converting our linked lists and hash tables to immutable stacks and immutable style and immutable red-black trees, we had no more problems with thread safety or performance. The resulting code was immaculate and surprisingly readable.
Frequently, a business rules engine requires you to roll your own state machine, which is very naturally modelled as a graph where vertexes and states and edges are transitions between states.
If for no other reasons, I'm glad I took the time to readable about data structures and algorithms simply to be able picture novel problems a little differently, especially combinatorial problems and graph problems. Graph theory is no longer a synonym for "scary".
What is the best way to create the best pseudo-random number generator? (any language works)
Best way to create one is to not to.
Pseudo-random number generators are a very complex subject, so it's better off to use the implementations produced by the people that have a good understanding of the subject.
It all depends on the application. The generator that creates the "most random" numbers might not be the fastest or most memory-efficient one, for example.
The Mersenne Twister algorithm is a popular, fairly fast pseudo-random number generator that produces quite good results. It has a humongously large period, but also a relatively humongous state (2.5 kB). However it is not deemed good enough for cryptographic applications.
Update: Since this answer was written, the PCG family of algorithms was published that seems to outperform existing non-cryptographic algorithms on most fronts (speed, memory, randomness and period), making it an excellent all-round choice for anything but cryptography.
If you're doing crypto though, my answer remains: don't roll your own.
The German magazine C't tested a number of software and hardware generators in the 2/2009 issue and ran the results through various statistical tests.
I scanned the results here.
I would not bother writing my own. The article mentions that even Donald Knuth failed with his "Super-random number generator", which was not so random after all. Get one that passed all tests (had a result > 0 in all columns). They also tested a setup with a VIA EPIA M10000 mobo, which has a hardware RNG. I like this option for a commercial or semi-commercial setup that requires a robust random number server with high throughput.
Unless, of course, you are just playing around, in which case this may be good enough.
PRNG algorithms are complicated, as is acquiring the right sources of entropy to make them work well. This is not something you want to do yourself. Every modern language has a PRNG library that will almost certainly be suitable for your use.
Yikes, that can get VEEEEEERY complicated! There seem to be a number of metrics for how to measure the "randomness" of a random number generator, so it's difficult to meaure which are "best". I would start with Numerical Recipes in C (or whatever langauge you can find one for) for a few examples. I coded up my first simple one from the examples given there.
EDIT: It's also important to start by determining how complex you need your random number generator to be. I remember a rude awakening I had in C years ago when I discovered that the default random number generator had a period somewhere around 32,767, meaning that it tended to repeat itself periodically after generating that many numbers! If you need a few dice rolls, that's fine. But not when you need to generate millions of "random" values for a simulation.
See Pitfalls in Random Number Generation
See this link for the TestU01 suite of tests, which includes several batteries of tests.
http://www.iro.umontreal.ca/~simardr/testu01/tu01.html
In the paper, the author demonstrates the test result on a variety of existing RNGs, but not .NET System.Random (as far as I can tell). Though he does test VB6's generator.
Very few pass all the tests...
Steal the one out of knuth seminumeric.
It is high quality and simple to implement.
It uses a pair of arrays, addition, and a couple of ifs.
Cheap, effective, and a nice long period 2^55 if i recall correctly.
If you're going to work in C++, Boost has a collection of PRNGs that I'd trust a lot more than whatever comes in standard libraries. The documentation might be helpful in picking one out. As always, how good a PRNG is depends on what you're using it for.
https://github.com/fsssosei/Pure_PRNG
Python libraries for PRNG algorithms that pass statistical tests
In production code it is definitely safer to leverage an established library, but understanding how pseudorandom number generators work and writing your own can be fun and worthwhile from an educational standpoint.
There are many different techniques, but one straight forward approach is based on the logistic map. I.e.
x = r * x * (1 - x)
With the right value for r, values of x demonstrate chaotic and unpredictable behavior. There are pitfalls to be aware of which you can read about in the references below.
References
https://tim.cogan.dev/random-num/
https://scholar.google.com/scholar?hl=en&as_sdt=0%2C44&q=Logistic+map%3A+A+possible+random-number+generator&btnG=
My favorites are Hardware random number generators